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Level-dependent Sugeno integral Radko Mesiar, Andrea Mesiarov´a-Zem´ankov´a, and Khurshid Ahmad

Abstract—A new concept of level-dependent Sugeno integral is introduced and used to represent comonotone maxitive aggregation functions acting on a complete scale K. Relationship between level-dependent Sugeno integral and some other types of fuzzy integrals is shown and properties of level-dependent Sugeno integral are discussed. Several examples show that level-dependent Sugeno integral can have different aggregation attitude for lower input values than for high input values and thus overcome problems which arise while using Sugeno integral. Index Terms—comonotone maxitivity, level-dependent Sugeno integral, Sugeno integral, weak universal integral.

I. I NTRODUCTION

I

N multicriteria decision making, several types of aggregation functions [5] play an important role. Indeed, if alternatives under discussion are evaluated by a set of criteria {1, . . . , n} (i means in short i’th criterion), each of them being quantified by means of a complete ordinal scale K (i.e., K is a complete chain with minimal element 0K and maximal element 1K , for example, K = [0, 1], or K = {weak, medium, excellent}), then aggregation function can be understood as (normed) utility function. We identify alternatives and their score vectors, i.e., the set A of alternatives under discussion is a subset of K n . For any aggregation function F : K n −→ K, a preference structure ¹F on A is given by x ¹F y if and only if F (x) ≤ F (y), where ≤ is the linear order on the chain K. Properties of the preorder ¹F are heavily related to the properties of the applied aggregation function F . On real interval scales, such as [0, 1], [0, ∞], usually the additivity is required, and then the corresponding aggregation functions are related to the standard Lebesgue integral (the class of weighted arithmetic means). However, often a weaker requirement of the comonotone additivity is sufficient, see e.g., [28], and then the aggregation functions based on Choquet integral [7], [8], [10], [24] should be considered. Nevertheless, on discrete scales the above approach cannot be applied (for example, the arithmetic mean on a discrete scale {1, 2, . . . , 10} cannot be defined by means of Choquet integral; for a deeper discussion see [17]). Therefore, when working on a general scale, maxitivity of aggregation functions comes into the picture. The class of maxitive aggregation functions is rather poor, and we will recall it in Section III. R. Mesiar is with the Institute for Research and Application of Fuzzy Modelling, University of Ostrava, Czech Republic. A. Mesiarov´a-Zem´ankov´a is with the Department of Computer Science, Trinity College, Dublin, Ireland, and with the Mathematical Institute of SAS, Bratislava, Slovakia. K. Ahmad is with the Department of Computer Science, Trinity College, Dublin, Ireland.

More exhaustive is the class of comonotone maxitive aggregation function, which is also examined and characterized in Section III. In this section, we introduce level-dependent Sugeno integral as a generalization of the integral introduced by Sugeno in [31]. Recall that the common Sugeno integral can be introduced on any scale K. In Section IV, we examine the relations of the level-dependent Sugeno integral with several integrals recently discussed in [14], see also [15], including Sugeno, Shilkret, Sugeno-Weber integrals, among others. Some properties of the level-dependent Sugeno integral are introduced in Section V. In concluding remarks we comment on how some applications based on Sugeno integral can be improved by using the level-dependent Sugeno integral discussed in the paper. II. P RELIMINARIES Throughout this paper, let K be a fixed ordinal scale with top element 1 and bottom element 0 (we omit the notation 1K and 0K for the sake of simplicity). For a fixed number n of criteria, we denote their collection by X = {1, . . . , n}, and for any subset E ⊂ X and constant c ∈ K, c · 1E ∈ K n is a score vector assigning the score c to any criterion i ∈ E, and assigning 0 to any criterion i ∈ / E. Definition 1 Two score vectors x, y ∈ K n are comonotone if there is a permutation σ : X −→ X such that xσ(1) ≤ · · · ≤ xσ(n) and yσ(1) ≤ · · · ≤ yσ(n) . Proposition 1 Let x, y ∈ K n be given. For the next four claims, (i) x and y are comonotone (ii) if xi > xj for some i, j ∈ X then yi ≥ yj (iii) {{i ∈ X | xi ≥ t} | t ∈ K}∪{{i ∈ X | yi ≥ t} | t ∈ K} is a chain (iv) for all t ∈ K, {i ∈ X | xi ∨ yi ≥ t} ∈ {{i ∈ X | xi ≥ t}, {i ∈ X | yi ≥ t}} it holds (i) ⇔ (ii) ⇔ (iii) ⇒ (iv). Proof: The equivalence of (i), (ii) and (iii) can be found in [3]. Suppose that (iii) holds. Then since {{i ∈ X | xi ≥ t}} ∪ {{i ∈ X | yi ≥ t}} is a chain for all t ∈ K, i.e., either {i ∈ X | xi ≥ t} ⊆ {i ∈ X | yi ≥ t} or {i ∈ X | yi ≥ t} ⊆ {i ∈ X | xi ≥ t} we have {i ∈ X | xi ≥ t} ∪ {i ∈ X | yi ≥ t} ∈ {{i ∈ X | xi ≥ t}, {i ∈ X | yi ≥ t}} and the equality {i ∈ X | xi ∨ yi ≥ t} = {i ∈ X | xi ≥ t} ∪ {i ∈ X | yi ≥ t} imply the desired result. Remark 1 Note that the implication (iv) ⇒ (i) need not hold, in general. Indeed, if x ≥ y (i.e., xi ≥ yi for all i ∈ X) then obviously (iv) is true, independently of the comonotone relation of x and y being valid or not.

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Definition 2 A mapping F : K n −→ K is called an aggregation function whenever it is non-decreasing in each coordinate and F (1∅ ) = 0, F (1X ) = 1. Definition 3 An aggregation function F : K n −→ K is said to be maxitive (comonotone maxitive) whenever F (x ∨ y) = F (x) ∨ F (y) for all x, y ∈ K n (for all comonotone x, y ∈ K n ). A well-known class of comonotone maxitive aggregation functions is constructed by means of Sugeno integral [1], [3], [24], [31] and fuzzy measure on X. Definition 4 A mapping m : 2X −→ K is called a (Kvalued) fuzzy measure on X if it is monotone and m(∅) = 0. If m(X) = 1 then m is called a normed fuzzy measure. Note that for any aggregation function F : K n −→ K, the set function mF : 2X −→ K given by mF (A) = F (1A ) is a normed fuzzy measure on X. Moreover, for each t ∈ K, the set function mF,t : 2X −→ K given by mF,t (A) = F (t · 1A ) is a fuzzy measure on X (compare also [2]). Hence to each aggregation function F : K n −→ K we can assign a nondecreasing system MF = (mF,t )t∈K of fuzzy measures such that mF,1 is a normed fuzzy measure. The problem when MF determines F univocally is partially solved in the next section. Definition 5 Let m : 2X −→ K be a normed fuzzy measure on X. Sugeno integral Sm : K n −→ K is given by _ Sm (x) = (t ∧ m({i ∈ X | xi ≥ t})). (1) t∈K

It is evident that for any normed fuzzy measure m, Sugeno integral Sm is an aggregation function. We summarize its properties (for the proof, see [3], [18], [24]). Proposition 2 Let m : 2X −→ K be a normed fuzzy measure on X. Then Sugeno integral Sm : K n −→ K is an aggregation function which is comonotone maxitive, min-homogenous (i.e., Sm (x ∧ c · 1X ) = Sm (x) ∧ c for any x ∈ K n and c ∈ K) and idempotent (i.e., Sm (c · 1X ) = c for all c ∈ K). III. C OMONOTONE MAXITIVE AGGREGATION FUNCTIONS AND THE LEVEL - DEPENDENT S UGENO INTEGRAL As already mentioned, to each aggregation function F : K n −→ K, a non-decreasing system MF = (mF,t )t∈K of fuzzy measures on X can be assigned, where mF,1 is a normed fuzzy measure. Example 1 For K = [0, 1] and n = 2, let F, G : [0, 1]2 −→ 2 [0, 1] be given by F (x1 , x2 ) = x1 +x and G(x1 , x2 ) = (x1 ∧ 2 2 . Then both F and G are aggregation functions and x2 )∨ x1 ∨x 2 MF = MG = (mt )t∈[0,1] , where mt ({1}) = mt ({2}) = 2t and mt ({1, 2}) = t. The above example shows that two different aggregation functions may posses the same assigned system of fuzzy measures. Therefore, when knowing only MF , to reconstruct

F , some additional information is needed. In some cases, this information restricts also the structure of MF . For example, on K = [0, 1] scale, the comonotone additivity of F forces mF,t = t · mF,1 for all t ∈ [0, 1], i.e., only one (normed) fuzzy measure mF,1 is necessary to determine F (which is then Choquet integral F = ChmF,1 : [0, 1]n −→ [0, 1] given R1 by F (x) = mF,1 ({i ∈ X | xi ≥ t})dt, see [3], [8], [24], 0

[28]). Similarly, if there is a pseudo-addition ⊕ : K 2 −→ K, i.e., an associative symmetric aggregation function on K with neutral element 0, and F : K n −→ K is pseudo-additive, i.e., F (x ⊕ y) = F (x) ⊕ F (y), then each mF,t is necessarily pseudo-additive, mF,t (A∪B) = mF,t (A)⊕mF,t (B) whenever A ∩ B = ∅, and then F (x) =

n M

mF,xi ({i}).

(2)

i=1

W Observe that the maximum operator : K 2 −→ K is a pseudo-addition on any scale K, and thus the following result is an immediate consequence of (2). Proposition 3 An aggregation function F : K n −→ K is maxitive if and only if F (x) =

n _

fi (xi ),

(3)

i=1

where fi (xi ) = mF,xi ({i}), xi ∈ K, i ∈ X, i.e., fi : K −→ K and (fi )i∈X is a system of non-decreasing functions such n W the fi (1) = 1 and fi (0) = 0 for all i ∈ X. i=1

The main aim of this section is the characterization of comonotone maxitive aggregation functions. Example 2 Let X = {1, 2}, i.e., F = [0, 1]2 . Define I : [0, 1]2 −→ [0, 1] by ( £ £2 x ∧ y if (x, y) ∈ 0, 21 , I(x, y) = x ∨ y else, (see Figure 1). Then I is an idempotent aggregation function which is comonotone maxitive. However, I(0, 1) = 1 and I(0 ∧ 41 , 1 ∧ 14 ) = I(0, 41 ) = 0 6= 14 ∧ 1, i.e., I is not min-homogenous and thus it cannot be a Sugeno integral, see Proposition 2. Example 2 shows that while the class of comonotone additive aggregation functions (on [0, 1]n ) reduces to the Choquet integral-based aggregation, the class of comonotone maxitive aggregation functions (on K n for any scale K) is much more wider than the Sugeno integral-based aggregation. Theorem 1 Let an aggregation function F : K n −→ K be comonotone maxitive. Then _ F (x) = mF,t ({i ∈ X | xi ≥ t}). (4) t∈K

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Proposition 4 Let SM : K n −→ K be a level-dependent Sugeno integral. Then SM is an aggregation function. Let m be a normed fuzzy measure on X. Defining a system M = (mt )t∈K by mt = t ∧ m (i.e., mt (A) = t ∧ m(A) for each A ⊆ X), t ∈ K, formula (5) reduces to (1). Hence each Sugeno integral-based aggregation function Sm is covered by (5). Due to Theorem 1 we see that each comonotone maxitive aggregation function F can be represented as the leveldependent Sugeno integral with respect to the system MF , F = SMF (compare Example 1, where G = SM ). Note that we do not distinguish the systems (mF,t )t∈K and (mF,t )t∈K ∗ because of mF,0 ≡ 0 for any F . However, also the opposite implication is true.

x∨y 1 2

x∧y

1 2

Fig. 1.

W tj

Theorem 2 An aggregation function F : K n −→ K is comonotone maxitive if and only if it is a level-dependent Sugeno integral SM with respect to some system M .

Aggregation function I from Example 2.

Proof: For any x ∈ K, x =

W t∈K

t · 1{i∈X|xi ≥t} =

tj · 1{i∈X|xi ≥tj } , where t1 < · · · < tk and {t1 , . . . , tk } =

{x1 , . . . , xn }. Then {i ∈ X | xi ≥ t1 } ) · · · ) {i ∈ X | xi ≥ tk } and thus any pair of functions tj · 1{i∈X|xi ≥tj } and tr · 1{i∈X|xi ≥tr } , j, r ∈ {1, . . . , k}, is comonotone. Moreover, F (tj · 1{i∈X|xi ≥tj } ) = mF,tj ({i ∈ WX | xi ≥ tj }). If F is comonotone maxitive, then F (x) = F (tj · 1{i∈X|xi ≥tj } ) = W W tj mF,t ({i ∈ X | xi ≥ t}), mF,tj ({i ∈ X | xi ≥ tj } = tj

t∈K

where the last equality follows from the monotonicity of the system MF and the fact that {i ∈ X | xi ≥ t} is constant for t ∈ ]tj−1 , tj ], j = 1, . . . , k + 1, with convention t0 = 0 and tk+1 = 1. Remark 2 (i) Note that the representation (4) can be derived also from the results presented in [2]. (ii) Since Sugeno integral is comonotone maxitive, (4) covers each Sugeno integral-based aggregation function Sm . Based on Remark 2(ii) and (4), we introduce now the leveldependent Sugeno integral. Definition 6 Let M = (mt )t∈K ∗ be a system of fuzzy measures on X, where K ∗ = K \ {0}, such that m1 is a normed fuzzy measure. The function SM : K n −→ K given by _ SM (x) = mt ({i ∈ X | xi ≥ t}) (5) t∈K ∗

is called the level-dependent Sugeno integral (with respect to the system M ). Observe first that the system M need not be monotone, and still the level-dependent Sugeno integral SM is nondecreasing. Moreover, due to the fact that m1 (X) = 1, we have SM (1X ) = 1. On the other hand, SM (1∅ ) = 0 follows from mt (∅) = 0 for all t ∈ K ∗ . We summarize the above results in the next proposition.

Proof: It is enough to show the comonotone maxitivity of SM . However, for any comonotone x, y ∈ K n , due to Proposition 1 it holds mt ({i ∈ X | xi ∨ yi ≥ t}) = mt ({i ∈ XW| xi ≥ t})∨mt ({i ∈ X | yi ≥ t}), and W hence SM (x∨y) = mt ({i ∈ X | xi ≥ mt ({i ∈ X | xi ∨ yi ≥ t}) = ( t∈K ∗ t∈K ∗ W mt ({i ∈ X | yi ≥ t})) = SM (x) ∨ SM (y). t})) ∨ ( t∈K ∗

Example 3 (i) Let X = {1, 2} and M = (mt )t∈[0,1] be given by ( t ∧ m∗ if t < 21 , mt = t ∧ m∗ else, where m∗ is the strongest fuzzy measure on X and m∗ is the weakest fuzzy measure on X, i.e., ( 0 if A = ∅, ∗ m (A) = 1 else, ( 1 if A = X, m∗ (A) = 0 else. Then SM = I, where I is given in Example 2. Observe that SM is an associative binary function, and that its nary extension is just the level-dependent Sugeno integral on Xn = {1, . . . , n} related to the system M as given above (dealing with the strongest and the weakest fuzzy measures on Xn ). (ii) Let X = {1, 2} and M = (mt )t∈[0,1] be given by  1 ∗  t ∧ m if t ≤ 3 , 1 mt = t ∧ m∗ if 3 < t ≤ 23 ,   t ∧ m∗ else. Then SM = SM ∗ for M ∗ = (m∗t )t∈[0,1] , where   if A = ∅, 0 m∗t (A) = t ∧ 31 if ∅ 6= A 6= X,   t if A = X

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£ ¤ for t ∈ 0, 23 and m∗t = t ∧ m∗ else. Moreover, ( £ ¤2 med 13 (x, y) if(x, y)∈ 0, 23 , SM (x, y)= max(x, y) else is again an associative binary function. Here med 13 (x, y) = med(x, y, 13 ). Again as in the case (i), when working on Xn , the corresponding leveldependent Sugeno integral is the n-ary extension of this binary SM .

Proof: If SM = Sm for some normed fuzzy measure m, then m∗t = mSM ,t = mSm ,t = t ∧ m and m∗1 = 1 ∧ m = m, implying (6). On the other hand, supposing of (6), W the validity for any x ∈ K n it holds Sm∗1 (x) = (t ∧ m∗1 ({i ∈ X | t∈K W W xi ≥ t})) = (t ∧ mv ({i ∈ X | xi ≥ t})) = ∗ ∗ t∈K v∈K Ã ! W W W mv ({i ∈ X | xi ≥ t}) = m∗t ({i ∈ t∈K ∗

Remark 3 Our approach to integration based on a system of fuzzy measures M , as well as the related concept based on Choquet integral proposed in [11] is linked to the idea of extension of aggregation function acting on subdomains to an aggregation function on full domain K. Note that several other types of integral-based aggregation based on a (finite) system of fuzzy measures, such as the two-step integral from [9], the multi-step Choquet integral from [20], the twofold integral from [21] or multifold integral discussed in [22], are linked to the composition of aggregation functions, compare [4]. IV. L EVEL - DEPENDENT S UGENO INTEGRAL AND WEAK UNIVERSAL INTEGRALS

Integrals like Sugeno and Shilkret, for instance, are type of the so-called weak universal integrals, i.e., the weakest N universal integral based on a given pseudo-multiplication , see [14], [16]. In this section we show how is the level-dependent Sugeno integral related to the weak universal integrals. Due to Theorems 1 and 2 we have the following result.

t∈K ∗

v∈K ∗ , v≤t

X | xi ≥ t}) = SM ∗ (x) = SM (x), where the last equality follows from Proposition 5. Example 4 Let M = (mt )t∈[0,1] , where ( t ∧ m∗ if t ≤ 12 , mt = t ∧ m∗ else. Then

  max f SM (f ) = min f  1

if max f ≤ 12 , if min f > 12 , else.

2

We have m∗t = mt for t ≤

1 2

and for t >

  0 ∗ mt (A) = t  1 2

1 2

it is

if A = ∅, if A = X, else.

Proof: By Theorem 2, SM is comonotone maxitive and thus, due to (5), SM = SM ∗ , where M ∗ = (m∗t )t∈K ∗ , m∗t W = mSM ,t , i.e., for all A ⊆ X, m∗t (A) = SM (t · 1A ) = mv (A).

Note that the system M ∗ = (m∗t )t∈]0,1[ satisfies (6) and thus SM = Sm∗1 . Observe that if X = {1, . . . , n} then SM coincide with the nullnorm med 21 , see [5], which is an n-ary extension of an associative binary aggregation function med 12 (x, y) = med(x, y, 12 ). Moreover, SM = Sm is the Sugeno integral with respect to the normed fuzzy measure m given by   0 if A = ∅, m(A) = 1 if A = X,  1 else. 2

The fact that the equality SM1 = SM2 may occur also if M1 6= M2 excludes the possibility of reconstructing M when only SM is known. Nevertheless, we can always recover the canonical system MSM = M ∗ . As already illustrated in Examples 2 and 3, there are proper level-dependent Sugeno integrals SM differing from any Sugeno integral Sm . The following theorem characterizes all systems M = (mt )t∈K ∗ yielding Sugeno integral, i.e., such that SM = Sm for some normed fuzzy measure m.

Recently, Klement et al. introduced in [14] universal integrals on the scale [0, 1], based on some semicopula N : [0, 1]2 −→ [0, 1] being a binary aggregation function on [0, 1] with neutral element 1. The weakest integral N IN,m : [0, 1]n −→ [0, 1] based on a fixed semicopula is then given by _ O IN,m (x) = (t m({i ∈ X | xi ≥ t})), (7)

Proposition 5 For any system M = (mt )t∈K ∗ of fuzzy measures on X such that m1 is a normed fuzzy measure there is a non-decreasing system M ∗ = (m∗t )t∈K ∗ such that SM = SM ∗ .

v∈K ∗ , v≤t

t∈[0,1]

Theorem 3 Under notation as in Proposition 5, a leveldependent Sugeno integral SM with respect to a system M = (mt )t∈K ∗ is also Sugeno integral, SM = Sm∗1 , if and only if m∗t = t ∧ m∗1 for all t ∈ K ∗ , i.e., for all A ⊆ X and t ∈ K ∗ it holds _ _ mv (A) = (t ∧ mv (A)). (6) v∈K ∗ , v≤t

v∈K ∗

where m is a given normed fuzzy measure on X (compare also with generalized fuzzy integrals introduced and discussed in [26]). Integrals IN,m are called weak universal integrals. ObserveN that Sugeno integral is related to the strongest semiN copula = ∧, Sm = I∧,m . Moreover, if = · is the standard product, I·,m is Shilkret integral [25], [29], while IT,m with T a t-norm is Sugeno-Weber integral [34].

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Theorem 4 Let IN,m : [0, 1]n −→ [0, 1] be a weak universal integral. Then IN,m = SMN,m is the level-dependent Sugeno N integral with respect to the system MN,m = (t m)t∈]0,1] .

Example 5 For K = [0, 1] and n = 2, define M = 2 (mt )t∈]0,1] by mt ({1}) = mt ({2}) = t2 and mt (X) = t. Then SM : [0, 1]2 −→ [0, 1] is given by

The proof follows directly when comparing (5) and (7). Corollary 1 Each weak universal integral IN,m is comonotone maxitive. Proof: The result is consequence of Theorems 2 and 4. V. S OME PROPERTIES OF LEVEL - DEPENDENT S UGENO INTEGRALS

Directly from Definition 6 follows the maxitivity of leveldependent Sugeno integral with respect to measure systems, (1) SM1 ∨M2 = SM1 ∨ SM2 , where for M1 = (mt )t∈K ∗ and (2) (1) (2) M2 = (mt )t∈K ∗ we have M1 ∨ M2 = (mt ∨ mt )t∈K ∗ . Consequently, SM1 ≤ SM2 whenever M1 ≤ M2 , i.e., if M1 ∨ M2 = M2 . Observing that m ≡ 0 is the weakest fuzzy measure on X, while m∗ is the weakest normed fuzzy measure on X, it is evident that the weakest system Mw related to a given scale K and given set X = {1, . . . , n} is given by Mw = (mt )t∈K ∗ , where m1 = m∗ , and mt ≡ 0 for t < 1. Therefore, the weakest level-dependent Sugeno integral SMw : [0, 1]n −→ [0, 1] is given by ( 1 if x = (1, . . . , 1), SMw (x) = 0 else, i.e., SMw = Fw is the weakest n-ary aggregation function on K. Similarly, the strongest fuzzy measure on X is m∗ (which is also normed), and thus the strongest system Ms = (m∗ )t∈K ∗ , induces the strongest level-dependent Sugeno integral SMs : [0, 1]n −→ [0, 1] given by ( 0 if x = (0, . . . , 0), SMs (x) = 1 else. Moreover, SMs = Fs is the strongest n-ary aggregation function on K. While Sugeno integral is an idempotent aggregation function, this is not the case of the level-dependent Sugeno integral, in general. In the following proposition we characterize all idempotent level-dependent Sugeno integrals. Proposition 6 The level-dependent Sugeno integral SM : K n −→ K with respect to a system of fuzzy measures M = (mt )t∈K ∗ is idempotent if and only if m∗t (X) = t for W each t ∈ K ∗ , i.e., mv (X) = t, t ∈ K ∗ . v∈K ∗ , v≤t

Proof: The necessity follows from the fact that _ m∗t (X) = SM (t · 1X ) = mv (X),

x2 y 2 ∨ . 2 2 is idempotent, however, it is not Sugeno integral. SM (x, y) = (x ∧ y) ∨

(8)

v∈K ∗ , v≤t

and the idempotency of SM ensuring SM (t · 1X ) = t. To see the sufficiency, it is enough to apply (8).

SM

The symmetry of an aggregation function F : K n −→ K means that F (x) = F (xσ ) for any permutation σ of X = {1, . . . , n}, where xσ = (xσ(1) , . . . , xσ(n) ). Using the notation of Proposition 6, we can characterize the symmetry of a leveldependent Sugeno integral SM as follows. level-dependent Sugeno integral Proposition 7 The SM : K n −→ K with respect to a system of fuzzy measures M = (mt )t∈K ∗ is symmetric if and only if for each t ∈ K ∗ , m∗t is a symmetric fuzzy measure, i.e., m∗t (A) = m∗t (B) whenever the sets A and B have the same cardinality. Proof: The necessity follows from the fact that SM (t · 1A ) = m∗t (A) and the sufficiency follows from Proposition 5. Recall that if we have monotone bijection ϕ : K −→ K (in decreasing case, ϕ is a negation on scale K; note that it need not exist, in general, take, e.g., K = {0, 1, . . . , ∞}), to each aggregation function F : K n −→ K one can assign its ϕ-transform F ϕ : K n −→ K, F ϕ (x1 , . . . , xn ) = ϕ−1 (F (ϕ(x1 ), . . . , ϕ(xn ))). For increasing ϕ, each ϕ-transform of a level-dependent Sugeno integral is again a level-dependent Sugeno integral, (SM )ϕ = SM ϕ , where if M = (mt )t∈K ∗ , then M ϕ = ϕ −1 (mϕ (mϕ(t) ). However, for decreast )t∈K ∗ , where mt = ϕ ϕ ing bijection ϕ, (SM ) need not be a level-dependent Sugeno integral. Note that in this case the comonotone maxitivity of SM is transformed into comonotone minitivity of (SM )ϕ , ϕ ϕ ϕ SM (x ∧ y) = SM (x) ∧ SM (y) whenever x and y are comonotone. It is an interesting open problem under which conditions on M , (SM )ϕ is a level-dependent Sugeno integral for any monotone bijection ϕ of the scale K. Recall that for Sugeno integral we have always (Sm )ϕ = Smϕ , where for A ⊆ X, mϕ (A) = ϕ−1 ◦ m(A) if ϕ is increasing and mϕ (A) = ϕ−1 (m(Ac )) if ϕ is decreasing. In the case of level-dependent Sugeno integral, it can be easily shown that for any decreasing bijection ϕ : K −→ K it holds (SMs )ϕ = SMw and (SMw )ϕ = SMs . Thus all constant systems, i.e., systems M = (mt )t∈K ∗ , with mt = m for all t ∈ K ∗ , and the extremal systems Ms and Mw solve our open problem. VI. C ONCLUSION We have represented all comonotone maxitive aggregation functions in the form of a level-dependent Sugeno integral. Level-dependent Sugeno integral can be applied on any scale K in any situation where Sugeno integrals were considered, extending significantly the discrimination power of the corresponding models, especially in multicriteria and decision aid. Recall that Sugeno integral was successfully used in a large

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variety of applications, among others in image processing, where both Sugeno and Choquet integrals have been evaluated in the creation of omnidirectional images - these images are produced by combining the output of a number of cameras at different vertical. A fuzzy version of set-theory based mathematical morphology was used to represent the intuitive concept of fuzzy edges of objects and arbitrary aspect of image partitioning [30], [33]. Sugeno integral has been used in financial forecasting [19] where it was also used for generating input vectors for functional-link perceptron and for training perceptrons in the system which predict bankruptcy [13]. Fuzzy integrals have been also used in speech recognition for example in classification of non-speech sounds [32] and for detection of syllables [6]. Medical applications based on Sugeno integral include endoscopic diagnosis introduced in [27]. The limitations of the Sugeno integral have been outlined in [23]. We have, furthermore, noted that in applications based on Sugeno integral very often the input-output relationship indicate that aggregation is related to Sugeno integral Sm for lower input values, while high values aggregation is related to Sugeno integral Sv , for some normed fuzzy measures m and v defined on the scale K = [0, 1]. In such a case, the definition of the level-dependent Sugeno integral SM based on the system M = (mt )t∈]0,1] given by mt = t ∧ m if t ≤ 21 and mt = t ∧ v else, yields an aggregation function coinciding with Sm whenever all inputs are below 12 , and it coincides with Sv if all inputs are above 12 . Note also that an alternative approach to level-dependent Sugeno and Choquet integrals based on ultrafilters was recently proposed by Havranov´a and Kalina in [12]. We believe that level-dependent Sugeno integral can improve the performance of a Sugeno integral, or other types of fuzzy integrals, in a range of applications mentioned above. Example 6 Assume the aggregation function I introduced in Example 2. For a normed fuzzy measure m on X = {1, 2} denote a = m({1}). Then Sm ((x, 0)) = min(x, a) and I((1, 0)) − Sm ((1, 0)) = 1 − a, while for x < 21 , Sm ((x, 0)) − I((x, 0)) = min(x, a). Thus for each m we have sup{|I(x)−Sm (x)| | x ∈ [0, 1]2 } ≥ max(1−a, min( 12 , a)) ≥ 1 2 , i.e., Sugeno integral cannot be used for approximation of I. However, due to Example 3 we see that I can be expressed as a level-dependent Sugeno integral. ACKNOWLEDGMENT This work was supported by grants MSM VZ 619889 8701, VEGA 2/71 42/27 and APVV-0071-06. The support of Trinity College Dublin is gratefully acknowledged. We are grateful to anonymous referees for several stimulating comments, and for Remark 1. R EFERENCES [1] P. Benvenuti, R. Mesiar, A note on Sugeno and Choquet integrals. In: Proc. IPMU’2000, Madrid (2000), pp. 582–585. [2] P. Benvenuti, D. Vivona, M. Divari, Aggregation operators and associated fuzzy measures. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 9 (2001), pp. 197–204.

[3] P. Benvenuti, R. Mesiar, D. Vivona, Monotone set functions-based integrals. In: E. Pap, editor, Handbook of Measure Theory, Vol II, Elsevier (2002), pp. 1329–1379. [4] T. Calvo, A. Mesiarov, L’. Val´asˇkov´a, Composition of aggregation operatorsone more new construction method. Kybernetika 39 (2003), pp. 643-650. [5] T. Calvo, A. Koles´arov´a, M. Komorn´ıkov´a, R. Mesiar, Aggregation Operators: Properties, Classes and Construction Methods. In: T. Calvo, G. Mayor, R. Mesiar, eds. Aggregation Operators. Physica-Verlag, Heidelberg (2002) pp. 3–107. [6] S. Chang, S. Greenberg, Syllable-proximity evaluation in automatic speech recognition using fuzzy measures and a fuzzy integral. In: Proc. 12th IEEE Fuzzy Systems Conference (2003), pp. 828–833. [7] G. Choquet, Theory of capacities. Annales de l’Institut Fourier 5 (19531954), pp. 131–295. [8] D. Denneberg, Non-additive Measure and Integral. Kluwer Acad. Publ., Dordrecht, (1994). [9] K. Fujimoto, T. Murofushi, M. Sugeno, Canonical hierarchical decomposition of Choquet integral over finite set with respect to null additive fuzzy measure. International Journal of Uncertainty, Fuzziness and KnowledgeBased Systems 6 (4) (1998), pp. 345–363. [10] M. Grabisch, T. Murofushi, M. Sugeno, Fuzzy Measures and Integrals. Physica-Verlag, Heidelberg, (2000). [11] S. Greco, S. Giove, B. Matarazzo, The Choquet integral with respect to a level dependent capacity. Fuzzy Sets and Systems, to appear. [12] Z. Havranov´a, M. Kalina, Fuzzy preference relations and Lukasiewicz filters. In: Proc. Eusflat 2007, Ostrava, vol. II (2007), pp. 337–341. [13] Y. Ch. Hu, F. M. Tseng, Functional-link net with fuzzy integral for bankruptcy prediction. Neurocomputing 70, (16-18) (2007), pp. 2959– 2968. [14] E. P. Klement, R. Mesiar, E. Pap, An universal integral. In: Proc. Eusflat 2007, Ostrava, vol. I (2007), pp. 253–256. [15] E. P. Klement, R. Mesiar, E. Pap, Integrals which can be defined on arbitrary measurable spaces. In: Proc. 28th Linz Seminar on Fuzzy Set Theory (Fuzzy Sets, Probability, and Statistics - Gaps ans Brigdes) (2007), pp. 72-77. [16] E. P. Klement, R. Mesiar, E. Pap, A universal integral based on measures of level sets. IEEE Transactions on Fuzzy Systems, submitted. [17] A. Koles´arov´a, G. Mayor, R. Mesiar, Weighted ordinal means. Information Sciences 177, (18) (2007), pp. 3822–3830. [18] J.-L. Marichal, On Sugeno integral as an aggregation function. Fuzzy Sets Systems 114 (3) (2000), pp. 347–365. [19] P. Melin, A. Mancilla, M. Lopez, O. Mendoza, A hybrid modular neural network architecture with fuzzy Sugeno integration for time series forecasting. Applied Soft Computing 7(4) (2007), pp. 1217–1226. [20] T. Murofushi,Y. Narukawa, A characterization of multi-step discrete Choquet integral. In: Proc. 6th International Conference on Fuzzy Sets Theory and its Applications (2002), p. 94. [21] Y. Narukawa, V. Torra, Twofold integral and multi-step Choquet integral. Kybernetika 40 (1) (2004), pp. 39-50. [22] Y. Narukawaa, V. Torra, Generalized transformed t-conorm integral and multifold integral. Fuzzy Sets and Systems 157 (2006), pp. 1384-1392. [23] H. T. Nguyen, V. Kreinovich, A modification of Sugeno integral describes stability and smoothness of fuzzy control. In Proc. IEEE Fuzzy Systems’1998, Anchorage, vol. 1 (1998), pp. 360–365. [24] E. Pap, Null-additive Set Functions. Kluwer Acad. Publ., Dordrecht (1995). [25] E. Pap, editor, Handbook of Measure Theory. Elsevier Science, Amsterdam (2002). [26] Z. Ruhuai, Generalized fuzzy integral. Journal of Fuzzy Mathematics 6 (1986), No.4, pp. 31–40. [27] K. Saito, K. Notomi, H. Hashimoto, M. Saito, Application of the Sugeno integral with λ-fuzzy measures to endoscopic diagnosis. Biomedical Soft Comuting and Human Sciences 9 (2003), pp. 11–16. [28] D. Schmeidler, Integral representation without additivity. Proceedings of the American Mathematical Society 97 (1986), pp. 255–261. [29] V. Shilkret, Maxitive measures and integration. Indagationes Mathematicae 33 (1971), pp. 109–116. [30] O. Strauss, F. Comby, Variable structuring element based fuzzy morphological operations for single viewpoint omnidirectional images. Pattern Recognition 40 (2007), pp. 3578–3596. [31] M. Sugeno, Theory of fuzzy integrals and applications. Ph.D. doctoral dissertation, Tokyo Institute of Technology (1974). [32] A. Temkoa, D. Macho, C. Nadeu, Fuzzy integral based information fusion for classification of highly confusable non-speech sounds. Pattern Recognition (doi: 10.1016/j.patcog.2007.10.026), (2007), 10 pp.

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[33] T. D. Pham, An image restoration by fusion. Pattern Recognition 34 (2001), pp. 2403–2411. [34] S. Weber, Two integrals and some modified versions: critical remarks. Fuzzy Sets and Systems 20 (1986), pp. 97–105.

Radko Mesiar received the Ph.D. Degree from Comenius University, Bratislava, Slovakia, and the D.Sc. degree from the Czech Academy of Sciences, Prague, in 1979 and 1996, respectively. He is a Professor of mathematics at Slovak University of Technology, Bratislava. His major research interests are in the area of uncertainty modeling, fuzzy logic and several types of aggregation techniques, nonadditive measures, and integral theory. He is coauthor of a monograph on triangular norms and author/coauthor of more than 100 journal papers and chapters in edited volumes. He is an Associate Editor of six international journals and a member of the European Association for Fuzzy Logic and Technology. He is a Fellow Researcher with UTIA AV CR Prague (since 1995) and IRAFM Ostrava (since 2005).

Andrea Mesiarov´a-Zem´ankov´a graduated at Faculty of Mathematics, Physics and Informatics of the Comenius University Bratislava. She obtained her Ph.D. at the Mathematical Institute of the Slovak Academy of Sciences in 2005, where she works as a researcher since then. Currently she is on the leave and works at Trinity College Dublin. Her major scientific interests are triangular norms, aggregation operators and fuzzy systems.

Khurshid Ahmad received his Masters in Physics from the University of Karachi, Pakistan with distinction in 1969 and PhD in Theoretical Nuclear Physics from the University of Surrey, England in 1974. He was a Lecturer in Physics at Karachi (1970-71) and a Research Fellow in Physics at Surrey (1974-1977). He joined Computing Unit at Surrey as Research Assistant in 1977 and became Group Leader in AI in 1985. He was appointed Senior Lecturer (1989) and Reader (1997) in the Department of Mathematics, Surrey. He was appointed Professor of Artificial Intelligence (1999); was the founding Head of the Department of Computing at Surrey (1999-2004); a Visiting Professor at the Copenhagen Business School (1997) and founded a technology start-up company, InKE in 1996. He has been the Professor of Computer Science at Trinity College (Univ. of Dublin), Ireland (2005). He has been a member of the UK EPSRC College of Computing (1999-to date) and a Chartered Engineer since 1987. He was elected to the Fellowship of the British Computer Society in 2005. He has published over 200 papers, two jointly authored books and four edited volumes, in computer-assisted learning, terminology and ontology, neural computing, fuzzy logic, information extraction, sentiment analysis and knowledge management. He has supervised 35 PhD students, managed and collaborated on 20 peer-reviewed projects in advanced IT, and led 6 technology transfer projects.

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